论文标题
在Navier-Stokes方程的一般简单网格上稳定稳定的Scott-Vogelius Pairs
Inf-sup stabilized Scott-Vogelius pairs on general simplicial grids for Navier-Stokes equations
论文作者
论文摘要
本文考虑了最近在[John/Li/Li/Merdon/Rui，Arxiv：2206.01242，20222]中引入了与时间相关的Navier-Stokes方程的离散化。在其中，速度空间是通过使用一些H（DIV）合并的Raviart-Thomas函数的H^1符合Lagrange元素空间获得的，从而确切地满足了差异约束。在这些方法中，可以使用任意形状的规范简单网格。 在本文中，考虑了两个用于离散对流术语的替代方案。一种变体导致一个仍然仅涉及音量积分的方案，而另一种变体则采用了从DG方案中已知的前风。两种变体都在某种意义上确保了线性动量和角动量的保护。此外，得出了压力射击和对流速度误差估计值，即，速度误差的结合不取决于压力和动能界限的误差的常数不会吹出微小的粘度。在浓缩未知数和所有非恒定压力未知数的凝结后，该方法可以减少为$ p_k-p_0 $的类似于任意速度多项式$ k $的系统。数值研究验证了理论发现。
This paper considers the discretization of the time-dependent Navier-Stokes equations with the family of inf-sup stabilized Scott-Vogelius pairs recently introduced in [John/Li/Merdon/Rui, arXiv:2206.01242, 2022] for the Stokes problem. Therein, the velocity space is obtained by enriching the H^1-conforming Lagrange element space with some H(div)-conforming Raviart-Thomas functions, such that the divergence constraint is satisfied exactly. In these methods arbitrary shape-regular simplicial grids can be used. In the present paper two alternatives for discretizing the convective terms are considered. One variant leads to a scheme that still only involves volume integrals, and the other variant employs upwinding known from DG schemes. Both variants ensure the conservation of linear momentum and angular momentum in some suitable sense. In addition, a pressure-robust and convection-robust velocity error estimate is derived, i.e., the velocity error bound does not depend on the pressure and the constant in the error bound for the kinetic energy does not blow up for small viscosity. After condensation of the enrichment unknowns and all non-constant pressure unknowns, the method can be reduced to a $P_k-P_0$-like system for arbitrary velocity polynomial degree $k$. Numerical studies verify the theoretical findings.
